CE262-Final Exam-Fall07

CE262-Final Exam-Fall07 - CE262 ENGINEERING DATA ANALYSIS...

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CE262 ENGINEERING DATA ANALYSIS Mark Hansen, Instructor FINAL EXAMINATION Dec. 17, 2007 NAME: ____________________________________________ Work on all six problems. Write clearly and state any assumptions you make. Show what you know—partial credit is generously given.
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Problem #1 Consider the following distribution for a continuous random variable: ( ) 0 ( ) 0 otherwise X X K f x K x x f x α α = - = a) Sketch this PDF. b) What is the CDF for X? c) Write an expression for K as a function of α . d) What is ( ) E X as a function of α ? e) What is ( ) VAR X as a function of α ? f) Suppose we have a set of observations, 1 2 , ,... n X X X drawn from this distribution. What is the method of moments estimator for α ? g) Suppose we have a set of observations, 1 2 , ,... n X X X drawn from this distribution. Write the likelihood and log likelihood functions that you would use to estimate α . h) Suppose we take the mean of 100 observations drawn from this distribution. What is the approximate PDF of the result?
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Problem #2 Three loaded cargo containers on a particular container ship are selected at random and weighed. The weights obtained are 10000, 20000, and 30000 kg. If the weight of a loaded container on this ship is assumed to be normally distributed: a) On the basis of these data, what is the 95% confidence interval for the average weight of a loaded freight car, if the variance of the distribution is known to be 25 tons 2 ? (Note: 1 ton=1000 kg).
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